We interpret the analysis step of the EnKF as a problem of coupling
the forecast distribution \(\pi _{X}\)  with the filtering
distribution \(\pi _{X\vert Y}\) . That is, given samples
\(x^1,\ldots ,x^M\)  from \(\pi _{X}\)  and a likelihood function
\(\pi _{Y\vert X}\) , we seek a transformation \(T\)  that yields
samples \(\smash{T(x^1),\ldots ,T(x^M)}\)  from \(\pi _{X\vert Y}\) .
Couplings are not unique. We are particularly interested in
transformations that can be estimated efficiently without resorting to
importance sampling—perhaps using only convex optimization—and
that are easy to “localize” in high dimensions. We want to avoid
the use of weights and thus issues of particle degeneracy. Moreover,
the computation of the transformation should not become
increasingly challenging as the variance of the observation noise
decreases. The latter is a typical concern of filtering algorithms
that rely on some attributes—e.g., moments—of the bootstrap particle
filter approximation. This paper proposes two new algorithms, called the
stochastic and deterministic map filters. The former represents a
non-Gaussian generalization of the EnKF with “perturbed
observations” [1]}, while the latter is a
nonlinear extension of the square-root EnKF
[2]}.
